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Basically, what you're saying is that a homogeneous electric field $\vec{E}(\vec{r},t)=\cos \omega t \ \hat {z}$ in vacuum requires you to have a magnetic field that satisfies
$$
\nabla \times \mathbf{B} =\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t},
$$
giving you a non-homogeneous magneti...